Korteweg-de Vries equationon slowly varying bottomDissipation of the soliton solution

Giulia Spina

Prof.Alfred Osborne,

University of Turin, July 2007

Why KdV?

• Nonlinear equation• Exact solutions• Describes long nonlinear waves in weakly

dispersive media:•Surface water waves

•Internal water waves

•Plasma physics

•Acoustic waves in crystal lattice

Purposes• Aim of the present work is to investigate the behavior of

the solitary wave (Tsunami) when the sea depth is not constant, say the sea bottom has a periodic profile.

In this presentation will be given hints on:

1. The Korteweg de Vries equation

2. The solitary wave on flat bottom

3. Discoveries on solitary wave on varying bottom

And results of the numerical simulation will be presented.

Long surface water waves in shallow water

• a is the wave amplitude

• h is the mean depth

• l is the wave length

• With these assumptions, the Laplace and Euler equations reads ( is the water potential, the water velocity)

y1)/(,/ 2 lhha

xx yy 0

y 0

t xx 1

y 0

t 2x

2 2y

2 0

0 y 1y 0

y 1

),(),( txutx

x

),( tx

)(xb

h

al

Theory

The Korteweg and De Vries equation (1895)

• perturbation expansion of as power series of leads to

nonlinear term dispersive term• Periodic solutions: cnoidal waves (one of the 12 Jacobi elliptic function)

depends on both and m

• Two limit cases: sinusoidal wave (m=0) and solitary wave (m=1)

One period of the cnoidal wave

062

30 xxxxxt uuuucu

a /h, (h / l)2

t 1

1 msin2()d

0

cn(t)cos()

Theory

solitary wave

• Exhibits discrete-object behavior

• Propagates without modifications on constant bottom

• Characteristic wave shape

• Velocity proportional

to the amplitude

)(

)4/(3

2

))((cosh),(

00

30

2)()(

02

00

ahgc

hak

eea

tcxkatxu

tcxktcxk

Theory

Zabusky and Kruskal’s numerical experiment (1965)

• Finite difference method to implement KDV

• Initial condition: sine wave of amplitude 1 cm propagating over 5 cm depth

RESULTS:• The asymptotic state of a (big enough) initial wave governed by the

KDV is formed of one or more solitons plus, eventually, radiation• The solitary waves retain their identity when they meet each other, apart

form a phase shift. • When two solitons of different amplitude meet, the smaller one is

negative shifted, the bigger one positive shifted

02 xxxxt uuuu

Theory

12

“The amplitude of oscillations grow and finally each oscillation achieves an almost steady amplitude and has a shape almost identical to that of an individual solitary-wave solution of KdV”

The wave steepens because of the

nonlinear term in regions were it has negative slope; the

dissipation term preserves from

breaking and oscillations of small wavelength develop

on the left of the front

Some special features of KdV

• Infinite number of conserved fluxes (found by Miura, Gardner and Kruskal, 1968)

...

0...)4

9

2

1()

2

1(

0)2

1(

32

1

2

1)

2

1(

0)64

3()(

4230

23

2320

2

20

xxtx

xxxxt

xxxt

uuucuu

uuuuucu

uuucu

Theory

062

30 xxxxxt uuuucu

Multiply KdV for

Energy conservation law

Mass conservation law

Multiply KdV for

No physical significance

u2

1

)3( 2xxuu

Some special features of KdV:

• Inverse Scattering Transform this solving method rises from Quantum Mechanics and is a general method

for finding the shape of an unknown potential by observing its Scattering Data, say the coefficient of reflection and transmission and its bound states.

In the context of KdV, the initial shape of the wave plays the role of the potential. Then one calculates the Scattering Data.

It is found that when the potential evolves with time according to KdV, the time evolution of the scattering data is trivial, so that one is able to calculate the Scattering Data at a given time t, and reconstruct the potential (say, the wave shape) at time t.

0)),(( txuxx

Incident waveReflected wave

Transmitted wave

Bound statesPotential

discrete eingenvalues are constant with time and correspond to the soliton component of the spectrum,

the reflection coefficient represents the radiation (oscillatory component of the spectrum)

tiebtb

dt

d

3

)0,(),(

02

0

0

Soliton climbing a shelf: fission Soliton in deeper water: radiation emission

Both phenomena arise by altering the relation between depth and amplitude

• Madsen and Mei (1969) – discovery by numerical simulation• Tappert and Zabusky (1971) – explanation based on Inverse Scattering

Theory: as the depth decreases the potential will “appear” deeper and more bound states will be possible.

• The initial sech^2 potential is reflectionless, but by changing its shape a reflection coefficient may arise, leading to radiation.

Theory

Fourier Transform

Theory

Two soliton appear as the depth

decreases from 10 to 5 cm

Radiation emission with increasing depth

Theory

• With varying bottom, the energy conservation law and the mass conservation law cannot be simultaneously satisfied. (Newell)

As a result, a trailing shelf appears behind the soliton.

Theory

• Constant adiabatic forcing due to raising bottom leads to non adiabatic transformation, i.e. fission, of the water solitary wave, could it lead to damping or forcing of the solitary wave mode?

More studies on varying bottom: Ono vs Ko and Kuehl

• Kakutani (1970), Ono (1972), Newell (1985)• assuming slowly varying bottom, the perturbation theory, with modified

bottom condition leads to

say,

• The conclusion: any change in mass and energy of the wave is determined by the initial and final depth. In the case of a inhomogeneous region surrounded by homogeneous ones, “The soliton propagates […] as if there were no inhomogeneous region”

note: this extra term can be absorbed by a coordinate change, leading to the KDV with varying coefficients (Djordjevc Redekopp,1978)

)(,)( xbyxb xxY

0))(ln(4

96 uD

d

duuuu

Theory

Potential term + derivative of the bottom function

04

96 u

D

Duuuu

Stretched coordinates:

Time

Space

An improved solution of the slowly variable coefficients KDV

• Ko and Kuehl (1978) noticed that, if one assumes that

in the linear limit, the perturbation expansion is valid even in the case we assume where is an arbitrary function of T.

Then the solution at first order of the perturbation expansion is

s=soliton, d=soliton distortion

and the first term doesn’t vanish when the bottom ceases to vary

• “The soliton experiences an irreversible loss of energy whenever it travels in a slowly varying medium”

• Necessary conditions are that the medium must vary on a scale long compared to that of which the soliton varies and that the fractional energy loss is small.

),,(),( Tutxu

)),~

((cosh 2)0( bau

Theory

More studies on varying bottom: Ono vs Ko and Kuehl

0)()( xxxxt uTuuTu tT

tx

~

ds uuuuu 10

Recent studies

• Grimshaw (2005)

asymptotic derivation of soliton amplitude decrease due to upward and downward long steps

• Agnon (1998)

“the cnoidal structure of the propagating nonlinear wave is destroyed if the topography contains a periodic component with a characteristic scale close to the nonlinearity length” the waves lose their spatially periodic structure.

Theory

Variable bottom KdV linearized

(damping or forcing)

• Nonlinear coefficient =0

damping or forcing depends on the sign of bottom function derivative

• Zero nonlinearity point: polarity inversion for internal waves. Internal solitons propagate on the interface between deep, heavy water layer

and the surface layer of lower density. When approaching a shelf the quotient between the two is reverted. As a consequence the incoming wave is damped and a solitary wave of opposite polarity raises (in Talipova et al., 1997 the origin of this process is identified in the trailing shelf) BUT in such a case other terms, i.e. , must be taken into account.

0 ubuu xxxxt

Theory

xxu xuu2

Varying bottom KDV and numerical implementation

• Dimensional form

• First order approximation (same equation as Johnson and Kakutani)

• Same method as Zabusky and Kruskal (finite difference method)

• Fortran programme

0262

3 x

xxxxxt

h

Numerical simulation

Black line: initial condition

Light blue line: evolution in time

The soliton amplitude rises

Soliton tail

Soliton amplitude decreases

Sine wave bottom-9 cases

• I tested several sine wavenumbers (300,1500,7500 cm period), that satisfy the request of slowly changing depth compared to the soliton perturbation.

• Three different amplitudes of the bottom perturbation (0.2, 0.4, 0.8 cm, unperturbed depth 5 cm) were tested.

• Results are obtained for 5 millions iterations (circa 2.5 hours, 1.5 Km- wave group velocity is 15 cm/s)

• Long time effect: soliton amplitude damping

Numerical simulation

b(x) is a sin wave of period 300cm

Numerical simulation

Numerical simulation

b(x) is a sin wave of period 1500cm

Numerical simulation

b(x) is a sin wave of period 7500cm

Faster damping: amplitude of the perturbation 0.8 cm, wave number 300Numerical simulation

Soliton amplitude vs time – all 9

cases

Numerical simulation

Period 300cm, 1500 cm

, 7500cm

Black line: amplitude 0,2

cm

Light blue line:

amplitude 0,4 cm

Dark blue line:

amplitude 0,8 cm

Ordered by period

Numerical simulationA

mplitude 0.2 cm

, 0.4 cm, 0.8 cm

Black line: period 7500 cm

Green line: period 1500 cm

Pink line: period 300 cm

Ordered by amplitude

Test of the results: Fourier Transform of initial and final wave form

The initial and final wave were analyzed by Fourier Transform, in order to

control the possible growth of high wavenumber modes due to the

dissipation term.

On the left pictures the 0,2 cm amplitude cases (from top to bottom: period 300cm,

1500 cm, 7500 cm). It looks like the different bottom period excites different wavenumbers

in the spectrum.

Below the 0,8 cm amplitude, 7500cm period case. The difference with the case on the left

is only quantitative.

Test of the results: conserved fluxes

• KDV equation has an infinite number of conserved quantities• By renormalizing coordinates it is possible to transform the variable bottom KDV into

the variable coefficient KDV (Djordjevic and Redekopp, 1978)

• According to Zabusky and Kruskal, with such an algorithm up to the fifth quantity is preserved.

Summary of the results

• Amplitude decrease in all 9 sinusoidal cases

• Time of decay depends on both amplitude and wave number of the perturbation: bigger amplitude of the bottom perturbation and wavelength of the same order of magnitude of the soliton horizontal dimension leads to faster and bigger damping.

• Forcing of wave numbers in the final wave is not random, but depends on the perturbation

Numerical simulation

Future developments

• The soliton amplitude reaches a mean asymptotic value, that depends just on the amplitude of the bottom perturbation?

• Analysis in terms of Riemann Theta functions (tool for the nonlinear analysis, analogue of the sine function for the Fourier Transform)

Acknowledgments

Many thanks to Professor Osborne for the continuous help and support and to Professors Onorato and Caselle for their kindness.

References• Agnon, Pelinowsky, Sheremet, “Disintegration of Cnoidal Waves over Smooth

Topography”, Studies in Applied Mathematics, 1998• Djordjevic Redekopp, “The Fission and Disintegration of Internal Solitary Waves

Moving over Two-Dimensional Topography”, Journal of Physical Oceanography, 1978• Grimshaw, Pelinowsky, Talipova, “Soliton dynamics in a strong periodic field: the

Korteweg-de Vries framework”, Physics Letters A, 2005• Madsen and Mei, “The Transformation of a solitary wave over an uneven bottom”,

Journal Fluid Mechanics, 1969• Kakutani, “Effect of Uneven Bottom on Gravity Waves”, Journal of the Physical Society

of Japan, 1971• Ko and Kuehl, “Korteweg-de Vries Soliton in a Slowly Varying Medium”, Physical

Review Letters, 1978• Miura, Gardner and Kruskal, “Korteweg-de Vries Equation and Generalizations.

Existence of Conservation Laws and Constants of Motion”, Journal of Mathematical Physics, 1968

• Newell, “Solitons in mathematics and physics”, SIAM, 1985• Ono, “Wave propagation in an Inhomogeneous Anharmonic Lattice”, Journal of the

Physical Society of Japan, 1972• Tappert and Zabusky, “Gradient-Induced Fission of Solitons”, Physical Review Letters,

1971• Zabusky and Kruskal, “Interaction of “Solitons” in a Collisionless Plasma and the

Recurrence of Initial States”, Physical Review Letters, 1965